Attitude Parameters

This paper makes use of three different attitude parameters to specify the orientation of a body (B) relative to another frame (such as the N-frame).

1. Direction Cosine Matrices
2. Quaternion Elements
3. Euler Angles

Direction Cosine Matrices

The first of these, the direction cosine matrix[1], \({^N}{R}{^B}\), specifies the relationship of one frame relative to another by relaying how the basis-vectors of one frame relate to the basis-vectors of another. These matrices have the property that they can, in a straightforward manner, transform vectors from one frame into another, such as from the Body to the NED-frame:

\[\vec{x}{^N} = {^N}{R}{^B} \cdot \vec{x}{^B}\]

In the upcoming derivation, transformations based on the Body-Fixed 3-2-1 Rotation set[2] and the formulation used by Thomas Kane[3] are relied upon extensively.

Quaternion Elements

The second parameter used to convey orientation information are quaternion elements[4] (also called Euler parameters), \({^N}{\vec{q}}{^B}\). Quaternions are relatively easy to propagate in time and do not possess singularities. However, they are not intuitive. Quaternions consist of a scalar and a vector component:

\[\begin{split}{^N}{\vec{q}}{^B} &= { \begin{bmatrix} { q_{0} \hspace{5mm} \vec{q}_{v} } \end{bmatrix} }^{T} \\ {\hspace{5mm}} \\ &= { \begin{bmatrix} {q_{0} \hspace{5mm} q_{1} \hspace{5mm} q_{2} \hspace{5mm} q_{3}} \end{bmatrix} }^{T} \\ {\hspace{5mm}} \\ &= { \begin{bmatrix} { \cos{\begin{pmatrix} \theta \over 2 \end{pmatrix}} \hspace{5mm} \hat{u} \cdot \sin{\begin{pmatrix} \theta \over 2 \end{pmatrix}} } \end{bmatrix} }^{T}\end{split}\]

Euler Angles

The final parameter used to relay attitude information are Euler angles. These are more intuitive than quaternions but, unlike quaternions, experience singularities at certain angles (based on the selected rotation sequence). For a 321-rotation sequence[5], the singularity occurs at a pitch of 90°.

Mathematical Relationships between Attitude Parameters

All three parameters contain the same information. The equations that relate the various parameters follow[6]. For a 321-rotation sequence, the expression relating the rotation transformation matrix of the body-frame in the NED-frame, \({^N}{R}{^B}\) , to the quaternion elements, \({^N}{\vec{q}}{^B}\), is:

\[{{^N}{R}{^B}} = { \begin{bmatrix} { \begin{array}{ccc} {{q_0}^2 + {q_1}^2 - {q_2}^2 - {q_3}^2} & {2 \cdot { \begin{pmatrix} {q_1 \cdot q_2 - q_0 \cdot q_3} \end{pmatrix} }} & {2 \cdot { \begin{pmatrix} {q_1 \cdot q_3 + q_0 \cdot q_2} \end{pmatrix} }} \cr {2 \cdot { \begin{pmatrix} {q_1 \cdot q_2 + q_0 \cdot q_3} \end{pmatrix} }} & {{q_0}^2 - {q_1}^2 + {q_2}^2 - {q_3}^2} & {2 \cdot { \begin{pmatrix} {q_2 \cdot q_3 - q_0 \cdot q_1} \end{pmatrix} }} \cr {2 \cdot { \begin{pmatrix} {q_1 \cdot q_3 - q_0 \cdot q_2} \end{pmatrix} }} & {2 \cdot { \begin{pmatrix} {q_2 \cdot q_3 + q_0 \cdot q_1} \end{pmatrix} }} & {{q_0}^2 - {q_1}^2 - {q_2}^2 + {q_3}^2} \end{array} } \end{bmatrix} }\]

\({^N}{R}{^B}\) can also be expressed in terms of Euler-angles, \({{^N}{\vec{\Theta}}{^B}} = { \begin{bmatrix} { {{^\perp}{\phi}{^B }} \hspace{5mm} {{^\perp}{\theta}{^B }} \hspace{5mm} {{^N}{\psi}{^\perp}} } \end{bmatrix} }^{T}\):

\[{{^N}{R}{^B}} = { \begin{bmatrix} { \begin{array}{ccc} { \cos{\begin{pmatrix} {{^\perp}{\theta}{^B}} \end{pmatrix}} } & { -\sin{\begin{pmatrix} {{^N}{\psi}{^\perp}} \end{pmatrix}} } & { 0 } \cr { \sin{\begin{pmatrix} {{^N}{\psi}{^\perp}} \end{pmatrix}} } & { \cos{\begin{pmatrix} {{^N}{\psi}{^\perp}} \end{pmatrix}} } & {0} \cr {0} & {0} & {1} \end{array} } \end{bmatrix} } \cdot { \begin{bmatrix} { \begin{array}{ccc} { \cos{\begin{pmatrix} {{^\perp}{\theta}{^B}} \end{pmatrix}} } & { \sin{\begin{pmatrix} {{^\perp}{\theta}{^B}} \end{pmatrix}} \cdot \sin{\begin{pmatrix} {{^\perp}{\phi}{^B}} \end{pmatrix}} } & { \sin{\begin{pmatrix} {{^\perp}{\theta}{^B}} \end{pmatrix}} \cdot \cos{\begin{pmatrix} {{^\perp}{\phi}{^B}} \end{pmatrix}} } \cr { 0 } & { \cos{\begin{pmatrix} {{^\perp}{\phi}{^B}} \end{pmatrix}} } & { -\sin{\begin{pmatrix} {{^\perp}{\phi}{^B}} \end{pmatrix}} } \cr { -\sin{\begin{pmatrix} {{^\perp}{\theta}{^B}} \end{pmatrix}} } & { \cos{\begin{pmatrix} {{^\perp}{\theta}{^B}} \end{pmatrix}} \cdot \sin{\begin{pmatrix} {{^\perp}{\phi}{^B}} \end{pmatrix}} } & { \cos{\begin{pmatrix} {{^\perp}{\theta}{^B}} \end{pmatrix}} \cdot \cos{\begin{pmatrix} {{^\perp}{\phi}{^B}} \end{pmatrix}} } \end{array} } \end{bmatrix} }\]

In this case, \({^N}{R}{^B}\) is broken up into two sequential transformations, which separate the roll and pitch calculations from the heading (this method is used later to form attitude measurements from the accelerometer and magnetometer readings):

\[{{^N}{R}{^B}} = {{^N}{R}{^\perp}} \cdot {{^\perp}{R}{^B}}\]

Finally, Euler angles, \({^N}{\vec{\Theta}}{^B}\), can be expressed in terms of quaternion-elements, \({^N}{\vec{q}}{^B}\):

\[\begin{split}{^\perp}{\phi}{^B} &= {atan2}{ \begin{pmatrix} { 2 \cdot { \begin{pmatrix} {q_2 \cdot q_3 + q_0 \cdot q_1} \end{pmatrix} }, \hspace{2mm} {{q_0}^2 - {q_1}^2 - {q_2}^2 + {q_3}^2} } \end{pmatrix} } \\ {\hspace{5mm}} \\ {^\perp}{\theta}{^B} &= -{asin}{ \begin{pmatrix} { 2 \cdot { \begin{pmatrix} {q_1 \cdot q_3 - q_0 \cdot q_2} \end{pmatrix} } } \end{pmatrix} } \\ {\hspace{5mm}} \\ {^N}{\psi}{^\perp} &= {atan2}{ \begin{pmatrix} { 2 \cdot { \begin{pmatrix} {q_1 \cdot q_2 + q_0 \cdot q_3} \end{pmatrix} }, \hspace{2mm} {{q_0}^2 + {q_1}^2 - {q_2}^2 - {q_3}^2} } \end{pmatrix} }\end{split}\]

Note

Due to the way the roll and pitch are separated from the heading, the Euler angles, \({^\perp}{\phi}{^B}\), \({^\perp}{\theta}{^B}\), and \({^N}{\psi}{^\perp}\) are the same if written as \({^N}{\phi}{^B}\), \({^N}{\theta}{^B}\), and \({^N}{\psi}{^B}\).

Attitude Parameters Example

Using the direction cosine matrix formulation, the transformation to get from the body to inertial-frame (ECEF) in Figure 1 is composed of multiple transformations:

\[{^E}{R}{^B} = {^E}{R}{^N} \cdot {^N}{R}{^\perp} \cdot {^\perp}{R}{^B}\]

Each transformation describes how one coordinate frame is related to the next in the sequence of rotations.

1. \({^\perp}{R}{^B}\): Transformation from the (light-blue) body-frame to the (dark blue) local perpendicular-frame \((\perp)\)
2. \({^N}{R}{^\perp}\): Transformation from the (dark blue) \(\perp\)-frame to the (red) local NED-frame
3. \({^E}{R}{^N}\): Transformation from the (red) NED-frame to the ECEF-frame (ECEF-Frame not shown; black line are latitude and longitude lines). \({^E}{R}{^N}\) is based on the WGS84 model.

This notation not only makes the formulation easier by simplifying the full complexity of the transformation but it helps avoid confusion by explicitly specifying the frame used in each calculation.

Some additional information about these frames:

1. \({^E}{R}{^N}\), the transformation between the NED and Earth-frame (used in the INS formulation), is solely a function of ECEF location, \({^E}{R}{^N} = f({\vec{r}}{^E})\), and is based on the WGS84 model.
2. \({^N}{R}{^B}\), the transformation between the NED and body-frame is solely a function of the attitude of the body-frame (roll, pitch, and heading angles of the body) and can be measured by the local gravity and magnetic-field vectors (or GPS heading), \({^N}{R}{^B} = f({\vec{g}}, {\vec{b}})\)

[1]	Pronounced “R B-in-N” and refers to the orientation of the B-Frame in the N-Frame. Also referred to as a rotation transformation matrix.

[2]

A 3-2-1 rotation set defines the attitude of one set of basis-vectors (local-frame) relative to another by specifying the angles of rotation required to get from the inertial (N) to the local-frame (L). With the local and inertial-frames initially aligned, the rotations are performed in the following order: the first is about the local z-axis (3), followed by a rotation about the local y-axis (2), and finally by a rotation about the local x-axis (1). The resulting matrix, \({^N}{R}{^L}\) = \({R}_{321}\), is composed of column vectors formed from the xyz-axes of the local-frame coordinatized in the inertial-frame: \({^N}{R}{^L}\) = \(\begin{bmatrix} {{{\hat{x}_{L}}{^N}} \hspace{5mm} {{\hat{y}_{L}}{^N}} \hspace{5mm} {{\hat{z}_{L}}{^N}}} \end{bmatrix}\).

[3]	Kane, Thomas R.; Levinson, David A. (1985), Dynamics, Theory and Applications, McGraw-Hill series in mechanical engineering, McGraw Hill. Note: one main difference between Kane’s approach is the DCM is the transpose of the DCM of other formulations; I think Kane’s formulation is more intuitive.

[4]

Commonly referred to simply as “quaternion”. To make it easier to reference the elements in c, c++, and python, the first quaternion-element (the scalar component of the quaternion) will have the zero index and is expressed as \({q}_{0}=\cos \begin{pmatrix} \theta / 2 \end{pmatrix}\). The vector component of the quaternion, \({\vec{q}}_{v}=\hat{u} \cdot \sin \begin{pmatrix} \theta / 2 \end{pmatrix}\), occupies elements 2, 3, and 4.

[5]	The 321-rotation sequence is the only rotation sequence considered in this paper.

[6]	Based on unpublished notes by Keith Reckdahl (Direction Cosines, Rotations, and Quaternions); this paper follows Kane’s approach closely. Any reference on the subject will work.